Free Energy Subadditivity for Symmetric Random Hamiltonians

Abstract

We consider a random Hamiltonian $H:\Sigma \to \mathbb{R}$ defined on a compact space $\Sigma$ that admits a transitive action by a compact group $\mathcal{G}$. When the law of $H$ is $\mathcal{G}$-invariant, we show its expected free energy relative to the unique $\mathcal{G}$-invariant probability measure on $\Sigma$ obeys a subadditivity property in the law of $H$ itself. The bound is often tight for weak disorder and relates free energies at different temperatures when $H$ is a Gaussian process. Many examples are discussed including branching random walk, several spin glasses, random constraint satisfaction problems, and the random field Ising model. We also provide a generalization to quantum Hamiltonians with applications to the quantum SK and SYK models.

References in corpus (8)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• On the overlap in the multiple spherical SK models
• Thermodynamics and Universality for Mean Field Quantum Spin Glasses
• The Golden-Thompson inequality --- historical aspects and random matrix applications
• Disorder chaos in the spherical mean-field model
• Correlation Inequalities for Spin Glasses
• TAP equations for orthogonally invariant spin glasses at high temperature
• Existence of the free energy for heavy-tailed spin glasses